Journal of Algebraic Combinatorics 9 (1999), 61 94 c 1999 Kluwer Academic Publishers. Manufactured in The Netherlands. Dynkin Diagram Classification of λ-minuscule Bruhat Lattices and of d-complete Posets ROBERT A. PROCTOR rap@math.unc.edu Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599 Received March 28, 1997; Revised September 15, 1997 Abstract. d-complete posets are defined to be posets which satisfy certain local structural conditions. These posets play or conjecturally play several roles in algebraic combinatorics related to the notions of shapes, shifted shapes, plane partitions, and hook length posets. They also play several roles in Lie theory and algebraic geometry related to λ-minuscule elements and Bruhat distributive lattices for simply laced general Weyl or Coxeter groups, and to λ-minuscule Schubert varieties. This paper presents a classification of d-complete posets which is indexed by Dynkin diagrams. Keywords: d-complete poset, minuscule Weyl group element, reduced decomposition, Dynkin diagram 1. Introduction A poset is d-complete if it satisfies certain local structural conditions which are listed in Section 3. In this paper we describe all possible d-complete posets. This includes an explicit listing of all possible irreducible components of d-complete posets. Each such irreducible component is indexed by a connected Dynkin diagram which is embedded in the order (Hasse) diagram of the component. The next three paragraphs are addressed to readers familiar with Lie and Coxeter groups and should be skipped by other readers. Except for those paragraphs and Section 15, this paper may be read by anyone who is familiar with basic poset concepts. There are combinatorial motivations for studying d-complete posets which are independent of Lie theory, e.g., the hook length property. Let W be a simply laced general Weyl group, i.e., the Weyl group associated to some fixed simply laced Kac-Moody algebra. Let λ be a dominant integral weight. Dale Peterson defines w W to be λ-minuscule if there exists some decomposition s ik s i1 of w such that s i j (s i j 1 s i1 λ) = (s i j 1 s i1 λ) α i j for 1 j k, where α i is the simple root associated to s i. In the companion paper [7] to this paper, we showed that if w is a λ-minuscule element for a simply laced general Weyl group, then the initial interval [e,w] in either the weak or strong Bruhat order on W/W λ is a distributive lattice. Set L w := [e,w]. These are the λ-minuscule Bruhat lattices mentioned in the title of this paper. Any finite distributive lattice L is determined by its subposet P of join irreducible elements. In [7] we characterized the posets P which could arise as posets of join irreducibles of Supported in part by NSA Grant No. MDA 904-95-H-1018.
62 PROCTOR λ-minuscule Bruhat lattices L w. This characterization amounted to the satisfaction of several certain local structural conditions; we defined a poset to be d-complete if it satisfied all of those conditions. The present paper describes all possible d-complete posets. (However, the definition of d-complete used in this paper is the order dual of the definition given in [7]; the statements made above are made with respect to the [7] definition.) Using a couple of easy translation steps from [7], the description of all d-complete posets given in the present paper can be converted into a description of all possible λ-minuscule elements for simply laced general Weyl groups. Let W be an arbitrary Coxeter group. Stembridge has studied the elements w W for which the weak Bruhat interval [e,w] is a distributive lattice [14]. He showed that this property is equivalent to w being fully commutative, namely, any reduced decomposition for w can be converted into any other reduced decomposition for w using only relations of the form s i s j = s j s i. It can be seen that every λ-minuscule element w is fully commutative. Section 15 contains further remarks on some Weyl group implications of this paper; also consult the last section of [7] for comments on representation theoretic and geometric implications. In particular, the list of d-complete posets given in this paper index a family of particularly simple Schubert subvarieties of Kac-Moody flag manifolds, and it is hoped that each poset drawn in this paper will embody much useful geometric information for the corresponding variety. Also, the d-complete posets describe the structures of the minuscule portions of weight diagrams for integrable representations of simply laced Kac-Moody algebras. In [10], we re-constructed some special cases of a basis of Lakshmibai [5] for Demazure modules in a poset-theoretic setting. We showed that a colored poset has such a basis for a module associated to it if and only if it is a colored d-complete poset. This was the origin of the notion of d-complete poset. Proposition 8.6 of [7] implies that the notions of d-complete poset and of colored d-complete poset are equivalent. The introduction for a general audience begins here. Section 3 contains the definition of d-complete poset. In Section 4 we show how to decompose an arbitrary connected d-complete poset into a slant sum of irreducible components. In Sections 5 and 6 we derive several facts about irreducible components; the most important being that a combinatorially defined subposet of an irreducible component, its top tree, must be Yshaped. In Sections 9 13 we define 15 exhaustive classes of irreducible components and describe all of the members of each class. In 14 out of the 15 classes, the top tree of an irreducible component must be a Dynkin diagram of general type E. This listing of possibilities is summarized in Section 7. Combining the theorems of Sections 4, 5, and 7 gives the classification of d-complete posets. Shapes (Ferrers diagrams) and shifted shapes are diagrams upon which Young tableaux and plane partitions are defined. The boxes of a (shifted) shape may be viewed as the elements of a certain poset. Then shapes and shifted shapes constitute two particular infinite families of posets. These two families of posets essentially form our Classes 1 and 2 of irreducible components. Two other infinite families of posets which are important for this paper are defined in Sections 2 and 4: these consist of double-tailed diamonds and rooted trees. The double tailed diamond posets play a central role in the definition of
DYNKIN DIAGRAM CLASSIFICATION 63 d-complete poset. Rooted tree posets are the trivial d-complete posets in a certain sense. By introducing the notions of d-complete and slant sum, and by describing the irreducible components in Classes 3 15, this paper may be thought of as filling out the category of posets hinted at by shapes, shifted shapes, double-tailed diamonds, and rooted trees. Slant irreducible components may be thought of as being the result of weaving together many double-tailed diamonds in a certain fashion. General connected d-complete posets are obtained by combining slant irreducible components with the slant sum operation. A poset P is said to be hook length if its associated P-partition generating function factors in a certain nice fashion analogous to identities discovered by Euler and Stanley. Until recently, the only known infinite families of hook length posets were shapes [12], shifted shapes [4, 11], rooted trees [12], and double-tailed diamonds. With Dale Peterson, we recently have shown that any d-complete poset is a hook length poset, by combining facts from algebraic geometry and representation theory with the viewpoint of [7]. A corollary to this result is a generalization of the hook product formula for the number of standard Young tableaux on an ordinary shape to a product formula for the number of order extensions of any d-complete poset. This corollary can be viewed as a conversion of Dale Peterson s (long known) hook formula for the number of reduced decompositions of a λ-minuscule element into a combinatorial form analogous to the original Frame-Robinson-Thrall form. After reading the definition of d-complete poset in Section 3 (with references to Section 2 as needed), browsers should also glance at the definition of slant sum in Section 4. Next they should read the last third of Section 4 (following the Corollary), which contains the theorem for decomposing d-complete posets into their irreducible components. This theorem is illustrated by figure 3. Then they should skim the description of the classification of irreducible components presented in Section 7, while consulting Table 1 and figures 5.1 5.15. Except for Sections 1, 14, and 15, this paper is entirely self-contained. It is possible [7] to recast the λ-minuscule definition above as a mild modification of the naively expressible numbers game on the simple graph G. This game has been studied in several papers, including [1 3, 6]. The paper [1] quickly shifts to such a naive environment. If one preferred, one could regard the Dynkin diagram classification result of this paper as being for certain aspects of a certain numbers game. In [8] we found all Bruhat orders on parabolic quotients W J of finite Weyl groups which are distributive lattices. We called the associated posets of join irreducibles minuscule, since they were exactly the posets of join irreducibles for the distributive lattices arising as weight diagrams for minuscule representations. In Section 14 we use our classification result to show that a d-complete poset is order self-dual if and only if it is a minuscule poset. (The minuscule posets are the only known Gaussian posets [13, p. 288].) 2. Poset definitions Let P be a poset (finite partially ordered set). If x is covered by y, we write x y. The order diagram of P is the directed graph made with such edges. Two subsets of a poset are non-adjacent if they share no elements and if there are no edges joining an element in
64 PROCTOR one to an element in the other. A poset is connected if its order diagram is connected. Any poset can be expressed as a direct sum of non-adjacent connected subposets. An ideal of P is a subset I P such that y I and x y imply that x P. A filter of P is a subset F P such that x F and y x imply that y P. Ifx P, then the principal ideal (x) := {y:y x}. If x, y P, we define intervals [x, y] :={z:x z y} and [x, y) := {z:x z<y}. A chain of length n in P is a sequence of elements x 0 x 1 x n. The order dual poset P* ofpis defined on the same set of elements as P by: y x in P*ifx yin P. Let Q denote the set of integral points in the strict fourth quadrant of the plane. Its elements (i, j) will be coordinatized as in a matrix, so i 1 and j 1. We turn Q into a poset by: (i 1, j 1 ) (i 2, j 2 ) if i 1 i 2 and j 1 j 2.Ashape λ = (λ 1,λ 2,...)is a finite filter of Q with λ i elements of the form (i, j). Note that λ 1 λ 2 λ r >0 for some maximal r 0. The width of λ is λ 1 and the length of λ is r. Let O denote the octant subposet of Q formed by taking the weakly upper triangular portion of Q : (i, j) O if j i. Ashifted shape µ = (µ 1,µ 2,...)is a finite filter of Q with µ i elements of the form (i, j). Note that µ 1 >µ 2 > >µ r >0 for some maximal r 0. The width of µ is µ 1 and the length of µ is r. When depicting such posets with graph paper, the up direction is Northwest, not North as usual. Let (r, c) denote the set of shapes λ whose length does not exceed r and whose width does not exceed c. Let (r,c) denote the set of shifted shapes whose length does not exceed r and whose width does not exceed c. Let (r,c) M denote the maximal element of (r,c), namely, the shifted shape with row lengths c, c 1, c 2,...,c r +1. The set (r,c) is defined to consist of all filters of the order dual of (r,c) M. The order diagrams of the double-tailed diamond posets d k (1) are shown in figure 1 for k = 3, 4, and 5. For k 3, the double-tailed diamond poset d k (1) has 2k 2 elements, of which two are incomparable elements in the middle rank and k 2 apiece form chains above and below the two incomparable elements. (The poset d k (1) is the poset of join irreducibles of the Bruhat lattice D k (1) for the ω 1 representation of the simple Lie algebra of type D k. [8]) The k 2 elements above the two incomparable elements are called neck elements, and when k 4 all but the lowest of these are called strict neck elements. Figure 1.
DYNKIN DIAGRAM CLASSIFICATION 65 Let P be a poset. A subset {w, x, y, z} of P is a diamond if z covers x and y, and each of x and y cover w. The top and bottom of this diamond are z and w, and the sides are x and y. An interval [w, z] isad 3 -interval if it is a diamond {w, x, y, z} for some x and y, or in other words, if [w, z] = d 3 (1). More generally, for k 3, we say that an interval [w,z]isad k -interval if is isomorphic to d k (1). Ad 3-interval [w; x, y] consists of three elements x, y, and w such that x and y each cover w. Fork 4, we say that an interval [w, y] isad k -interval if is isomorphic to d k(1) {t}, where t is the maximal element of d k (1). 3. d-complete posets Let P be a poset with elements w, x, and y. Suppose that [w; x, y] isad 3 -interval. If there is no z P such that {w, x, y, z} is a d 3 -interval, then [w; x, y] isanincomplete d 3 -interval. If there exists w w such that [w ; x, y] is also a d 3 -interval, then we say that [w; x, y] and [w ; x, y] overlap. A poset P is d 3 -complete if it contains no incomplete d 3 -intervals, if the maximal element of each d 3-interval does not cover any elements outside of that interval, and if it contains no overlapping d 3 -intervals. We have just required: (D1) Anytime two elements x and y cover a third element w, there must exist a fourth element z which covers each of x and y, (D2) If {w, x, y, z} is a diamond in P, then z covers only x and y in P, and (D3) No two elements x and y can cover each of two other elements w and w. Let k 4. Suppose [w, y]isad k -interval in which x is the unique element covering w. If there is no z P covering y such that [w, z]isad k -interval, then [w, y]isanincomplete d k -interval. If there exists w w which is covered by x such that [w, y] is also a d k -interval, then we say that [w, y] and [w, y] overlap. For any k 4, a poset P is d k -complete if: (D4) There are no incomplete d k -intervals, (D5) If [w, z] isad k -interval, then z covers only one element in P, and (D6) There are no overlapping d k -intervals. A poset P is d-complete if it is d k -complete for every k 3. It is easy to see that any filter of a d-complete poset is itself d-complete. Note that D5 implies: Any strict neck element of a d k -interval (necessarily k 4) in a d-complete poset covers exactly one element. Now we list some properties which follow from D1 and D2. Proposition Let P be a poset with elements w, x, x, y, z which satisfies D1. Then the finiteness of P implies: (F1) Suppose that x x, x y, and y x. Then there exists a y P which covers x. (F2) If P is connected, then it has a unique maximal element z 0.
66 PROCTOR Figure 2. (F3) If P is connected, every chain from an element w to z 0 has the same length. If in addition P satisfies D2, then: (F4) If P is connected, each element of P other than z 0 is covered by 1 or 2 elements. Proof: For F1, let x =: x 0, x 1,..., x k := x be such that x i covers x i 1 for 0 < i k. Let y 0 := y. For 1 i k apply D1 to [x i 1 ; x i, y i 1 ] to see that a y i exists which covers x i and y i 1. It is never the case that y i = x, since y i y but x y. Let y := y k. For F2, note that P has at least one maximal element since it is finite. Suppose it has at least two maximal elements z 0 and z 1. Since P is connected, there will be a up/down path from z 0 to z 1 of shortest possible length. Let y be the earliest element such that y z 0 and let x be the preceding element on the path: x z 0. Applying F1 produces a contradiction of the maximality of z 0, and so z 0 must be unique. For F3, let w be maximal in P such that w a 1 a r = z 0 and w b 1 b s = z 0 with r > s. Clearly s > 1, and so b 1 < z 0. Use D1 to construct c 2, c 3,... along a 1 a r = z 0 such that a 1 c 2 and b 1 c 2, then a 2 c 3 and c 2 c 3, etc. By the maximality of z 0, there must exist some t r such that c t = a t. Thus b 1 c 2 c t = a t a r = z 0 and b 1 b 2 b s = z 0 are chains of lengths r 1 > s 1, contradicting the maximality of w. For F4, note that if an element is covered by three or more other elements, then D1 and D2 force P to be infinite, as illustrated in figure 2. 4. Slant sum decomposition It is easy to see that a poset is d-complete if and only if each of the posets defined by the connected components of its order diagram is d-complete. So for our classification, we will consider only connected d-complete posets. The notion of slant sum introduced here will be used to break connected d-complete posets into smaller pieces, as illustrated in figure 3.
DYNKIN DIAGRAM CLASSIFICATION 67 In this paper, a rooted tree is a poset which has a unique maximal element, and is such that each non-maximal element is covered by exactly one other element. It is easy to see that rooted tree posets are d-complete. Let P be a connected poset with a unique maximal element z. Atop tree element x P is an element which is covered by at most one other element and is such that every y x is covered by at most one other element. The top tree T of P consists of all top tree elements. It is easy to see that T is a filter of P which is a rooted tree under the order inherited from P. Obviously the top tree of a rooted tree T is all of T. Let P be a connected d-complete poset with top tree T. An element y P is acyclic if y T and it is not in the neck of any d k -interval for any k 3. An element of P is cyclic if it is not acyclic. If y P is cyclic, then either it is in the neck of some d k -interval or there exists some z y which is covered by two elements. If y T, then it is cyclic if and only if it is in the neck of some d k -interval. Let P 1 be a d-complete poset containing an acyclic element y. Let P 2 be a connected d-complete poset which is non-adjacent to P 1. By F2, let x denote the unique maximal element of P 2. Then the slant sum of P 1 with P 2 at y, denoted P 1 y \ x P 2, is the poset formed by creating a covering relation x y. Ad-complete poset P is slant irreducible if it is connected and it cannot be expressed as a slant sum of two non-empty d-complete posets. Suppose that P is a connected d-complete poset with top tree T. An edge x y of P is a slant edge if x, y T and y is acyclic. All of the elements of a rooted tree P 1 are acyclic. A one element poset P 2 is d-complete. If y P 1, then the slant sum P 1 y \ x P 2 will be a rooted tree. Any iterated slant sum formed with one element posets will be a rooted tree, and every rooted tree can be obtained in this fashion. All of the edges of a rooted tree are slant edges. One element posets are slant irreducible. Removing all of the edges of a rooted tree produces a disjoint union of slant irreducible posets. What if we remove a slant edge from an arbitrary connected d-complete poset? Proposition A Let P be a connected d-complete poset and let x y be a slant edge. Let P 2 := (x) and let P 1 := P (x). Then P is a slant sum P y 1 \ x P 2 of two non-adjacent connected d-complete posets P 1 and P 2. The top tree of T of P is the slant sum T y 1 \ x T 2 of the top trees T 1 and T 2 of P 1 and P 2. The acyclic elements of P are acyclic in P 1 or P 2, and slant edges other than x y are slant edges in P 1 or P 2. Proof: By definition, y is acyclic. Since y is the only element covering x in P, there cannot be any d k -conditions passing through x and beyond. Suppose that some element u x in (x) is covered by some element v in P (x). Since x is covered only by y, an argument similar to that used to prove F1 could be used to show that y would be the top element of a diamond. So aside from x y, the subsets P 1 and P 2 are non-adjacent and any d k -intervals contained in (x) P are completed within (x). Hence P 2 := (x) is d- complete. Since P 2 is an ideal of P, the subset P 1 is a filter. Hence P 1 is d-complete. Clearly y will remain acyclic in P 1. Obviously x is the maximal element of (x). Removing one
68 PROCTOR edge of a connected poset creates at most two components. So P = P y 1 \ x P 2 as claimed. The other statements are easy. Going in the other direction, we have: Proposition B Let P 1 be a connected d-complete poset with acyclic element y and let P 2 be a connected d-complete poset with maximal element x. Then the slant sum P := P y 1 \ x P 2 is a connected d-complete poset. If T 1 and T 2 are the top trees of P 1 and P 2, then T y 1 \ x T 2 is the top tree of P. The acyclic elements of P 1 and P 2 are acyclic in P, the slant edges of P 1 and P 2 are slant edges in P, and the edge x y is a slant edge in P. Proof: The added edge is the only edge joining an element of P 1 to an element of P 2. Since y is not below any element which is covered by two elements and x is not covered by any other element, adding a downward edge at y cannot cause a violation of D1, D4, D3, or D6. Since y is not in the neck of any d k -interval in P 1, adding this edge cannot cause a violation of D2 or D5. So P y 1 \ x P 2 is d-complete. It is given that y is acyclic in P 1 ;in order for y to not be acyclic in P, the element x would have to be the maximal element of some d k -interval in P 2 for some k 4. This is impossible since P 2 is d-complete. The other statements are easy. These two results can be combined to immediately yield: Proposition C Let P be a connected d-complete poset. Then P is slant irreducible if and only if it contains no slant edges. Also, P is slant irreducible if and only if every acyclic element is a minimal element of its top tree. We now use this proposition to describe the structure of any connected d-complete poset P. First locate all of its slant edges. These may be erased in any order to produce a collection P 1, P 2,...of uniquely determined smaller non-adjacent connected d-complete posets. No new slant edges are created, and so each of P 1, P 2,...are slant irreducible. We say that P 1, P 2,...are the slant irreducible components of P. Conversely, suppose that P 1, P 2,...are slant irreducible d-complete posets in which we have identified all acyclic elements. In general, many different possible larger connected d-complete posets can be formed from these posets by forming various slant sums. Acyclic elements remain acylic and may be used more than once, but maximal elements cease to be maximal after being used as the bottom of a slant edge. This procedure generalizes the process of producing any rooted tree by forming an iterated slant sum of one element posets. There exist slant irreducible d-complete posets with no acyclic elements; given only such posets we would not be able to form larger connected d-complete posets. The remaining sections of this paper will be devoted to listing all possible slant irreducible d-complete posets. We will regard the one element poset as the trivial slant irreducible d- complete poset. By irreducible component we will mean a slant irreducible d-complete poset which has two or more elements. An irreducible component has a unique maximal
DYNKIN DIAGRAM CLASSIFICATION 69 Figure 3. element and is not a rooted tree. Hence its order diagram contains at least one cycle when viewed as a graph. Theorem Let P be a connected d-complete poset. It may be uniquely (up to the order of operations) decomposed into a slant sum of one element posets and irreducible components. The top tree of P is an analogous slant sum of the top trees of the irreducible components. In order to avoid producing a huge number of one element components during the decomposition, one might want to avoid erasing slant edges which occur within trees. It is possible to describe a smaller set of cut edges whose removal will produce a slant sum of maximal subtrees and (non-trivial) irreducible components: Suppose that P is a connected d-complete poset with top tree elements x and y. A slant edge x y is a down cut edge if y is the side element of some diamond in P;itisanup cut edge if x is cyclic. A slant edge may be both a down cut edge and an up cut edge. In figure 3, both kinds of cut edges are denoted with double slash marks. Erasing all of these edges produces our preferred grapevine view of an arbitrary connected d-complete poset: Such a poset consists of many irreducible components (viewed as bunches of grapes: any irreducible component contains at least one cycle; view the minimal cycles as grapes) which are connected together with maximal tree portions (portions of the vine outside of the bunches). The only botanically incorrect aspect of this model is that our vine can recontinue from a corner of a bunch of grapes. The larger dots and the heavier edges indicate the top trees of the irreducible components of this d-complete poset. 5. The top tree of an irreducible component From now on, let P denote some fixed irreducible component, namely, a slant irreducible d-complete poset with at least two elements. In this section we learn that the top tree T of P must be Y-shaped.
70 PROCTOR Figure 4. Lemma Let T be the top tree of an irreducible component P. Suppose z, y T are such that z is the top of a diamond with side element y. Then y is a diamond top in P if and only if it is not a minimal element of T. Proof: If y is not minimal in T, it is cyclic in P by Proposition 4.C. Since the bottom of the diamond whose top is z cannot be in T, we know that y covers at least two elements. So it cannot be cyclic by being a strict neck element of a d k -interval for some k 4. Hence y must be a diamond top. Suppose that y is a diamond top in P. If each of the elements covered by y is covered by some element different from y, there would be two distinct elements covering y. But y T. So one element covered by y is covered by only y, and that element must be in T, contradicting the minimality of y in T. We now need to define a particular kind of rooted tree. Let f 0 and h g 0 be integers. The rooted tree Y ( f ; g, h) consists of one branch element above which a chain of f elements has been adjoined and below which two non-adjacent chains with g and h elements, respectively, have been adjoined toward the left and toward the right, respectively. From now on, all order diagrams will be rotated 45 counterclockwise before being drawn, so that the up direction is Northwest. With this convention, the order diagram for Y ( f ; g, h) appears as the f + g + h + 1 topmost and leftmost elements in figure 4. Theorem Let P be an irreducible component. Then its top tree T is of the form Y( f ; g, h) for some f 0 and h g 1. Proof: Consider a non-minimal element z of T. By Proposition 4.C, it is cyclic. Since z T, it must be in the neck of some d k -interval for some k 4. But elements in the strict neck of a d k -interval for k 4 cover exactly one other element of P. Therefore z can cover two or more other elements of T only if it is the top of some diamond in P. The only way z can cover two or more other elements of T and be a diamond top in P is for the side elements x and y of the diamond to be the other elements of T covered by z. Of course z cannot cover any other elements of P, oroft. So branches in T can only be
DYNKIN DIAGRAM CLASSIFICATION 71 two-fold branches such that there exists a fourth element of P which is covered by each of the branching elements. Since P is not a tree, the set P T is non-empty. Let w be a maximal element of P T. Note that w must be covered by two elements of T, call them x and y. Then there exists an element z of T which covers both x and y. SoThas at least one branching. Now we show that there can only be one such branching. Let z be a minimal such branch node, which covers x and y. Let v be minimal in T such that v>zand such that v is another such branch node. Let t and u be covered by v, and let s T be covered by t and u. One of t and u must be above z in T ; suppose that it is u. By the lemma, u is a diamond top. One of the side elements of this diamond must be in T ; call it r. Let q be the bottom of this diamond. Application of the lemma can be iterated until an element analogous to r is actually z. For simplicity of notation, depict this with r = z. Then not only would z cover x and y as a diamond top, it would also cover q. This is impossible by D2. Therefore, we must have q coinciding with either x or y. But q T and both x, y T. Therefore, there is exactly one branch in T, and it is a two-fold branch. Draw P so that the longer lower branch of T is to the right. 6. The top filter of an irreducible component We continue to consider the fixed irreducible component P. Let f, g, and h be such that the top tree T of P is Y ( f ; g, h). Let e 0 be the branch node in T. Let e 1 and e 2 be the elements of T covered by e 0, with e 1 being on the left branch (the one of length g). Let e 3 T be the unique element of P covered by e 1 and e 2 whose existence was noted in the proof of the theorem. Successively label the elements of T above e 0 by n 1,...,n f. Label the remaining elements on the two lower branches by a 2,...,a g and d 2,...,d h. All of these elements named so far are shown in figure 4, as are some other elements whose existence will soon be proved. Given this notation, we can state some additional facts which are apparent from the proof of Theorem 5: Proposition A Let P be an irreducible component with top tree T = Y ( f ; g, h). For 2 i f, the element n i covers only n i 1.For1 i f 1, the element n i is covered only by n i+1. The element e 0 is covered only by n 1 and is a diamond top of a diamond with side elements e 1 and e 2. Acyclic elements of irreducible components can only possibly occur at two specific locations: Proposition B Let P be an irreducible component with top tree T = Y ( f ; g, h). The branch elements a 2,...,a g 1 and d 2,...,d h 1 are diamond tops. The minimal elements a g and d h of T cannot be diamond tops, and these are the only two elements of P which could be acyclic elements. Each of these elements is acyclic if and only if it is not in the strict neck of some d k -interval for some k 4. Proof: Continuing the proof of Theorem 5, repeated applications of Lemma 5 can be used to show that all but the last element of each lower branch are diamond tops. And
72 PROCTOR Lemma 5 implies that these last elements are not diamond tops, since they are minimal in T. These minimal elements of T are the only possible acyclic elements of P. Since they cannot be diamond tops, each fails to be cyclic precisely when it lies in the strict neck of some d k -interval for some k 4. Before proceeding, we need to state a local structural fact for d-complete posets. Lemma Let P be a d-complete poset, with z, x, y P. Suppose that z covers x and y, each of which is a diamond bottom. Then z can cover at most one other element q. This element q can be covered by no other elements, and any element covered by q can be covered only by q. Proof: Let r and s denote the diamond tops covering z which correspond to the diamond bottoms x and y. Let u denote the diamond top covering r and s. Ifzcovered both q 1 and q 2, then [q 1, u] and [q 2, u] would violate D6 with k = 4. Suppose q is covered by z. Ifq is also covered by an element other than z so that D3 is not violated, then a third diamond top t would cover z, violating F4. Let w be covered by q. Ifwis covered by an element v other than q, there must be a diamond top corresponding to the diamond bottom w. It cannot be z by D2 since z covers x and y. (This would be the case if v = x or v = y.) Hence, an element other than z covers q, violating an earlier conclusion. We return to the consideration of the fixed irreducible component P with top tree T = Y ( f ; g, h). Let b 2,...,b g and c 2,...,c h denote the bottoms of the diamonds whose existence arose during the proof of Proposition B. If f 0 and h g 1, define the poset U( f ; g, h) by the order diagram of figure 4. We now show that the irreducible component P must contain a filter of this form; we call it the top filter of P. Proposition C If the irreducible component P has top tree T = Y ( f ; g, h), then it contains a filter of the form U( f ; g, h), which itself contains T. The only elements of U( f ; g, h) which can cover elements outside of U( f ; g, h) are b 2,...,b g,c 2,...,c h, t 1,...,t f. Proof: The diamond top e 0 cannot be a strict neck element for any d k -interval. Therefore, in order for the edge e 0 n 1 to not be a slant edge, the element n 1 must be the top of some d 4 -interval with diamond bottom e 3. Hence e 3 covers some element t 1. Apply the lemma with z = e 3.Sot 1 is covered only by e 3, and e 3 covers no elements aside from b 2, c 2, and t 1. Repeating the slant edge reasoning for n i with i 2 leads to the existence of t 2,...,t f as shown. Upward diamond propagation implies that t i can be covered only by t i 1. By F4, none of b 2,...,b g or e 3 or c 2,...,c h can be covered by other elements. The diamond tops a 2,...,a g 1 and d 2,...,d h 1 cannot cover elements other than the two elements shown. If a g covers another x aside from b g, then x could be covered only by a g, or else upward propagation of diamonds would change T. But then x T, contradicting our assumed T.
DYNKIN DIAGRAM CLASSIFICATION 73 7. The list of possible irreducible components Recall that an irreducible component is a slant irreducible d-complete poset which contains two or more elements. In Sections 9 13, we will define 15 disjoint classes of irreducible components C 1,...,C 15 which will be seen to exhaust the set of all irreducible components. Here we present the resulting list, which is indexed by Table 1. For each triple of values f 0, g 1, and h g allowed by the Nth line of Table 1, define the maximal poset M N = M N ( f ; g, h) to be the poset defined by the order diagram of figure 5.N. (One is to take f, g, and h large solid dots, respectively, to the left, below, and right of the junction large solid dot.) Also define the minimal poset L N = L N ( f ; g, h) to be the filter of M N consisting of all large solid dots, circled dots, solid dots, solid squares, and boxed squares. (In other words, all elements of M N other than hollow dots and hollow triangles.) We will prove that the Nth class of posets C N consists of all posets which are filters of M N ( f ; g, h) containing L N ( f ; g, h), as f, g, and h run over all values allowed by the table. These posets will all be distinct, and so each irreducible component will occur exactly once in our listing. Table 1. Class Colloquially f g h Name λ µ Acylics 1 Shapes =0 1 g a n [0; g, h; λ] (g 1, h 1) L, R 2 Shifted shapes =1 = 1 1 d n [1; 1, h; µ] (h,h) 1 L 2,R 3 Birds 1 2 g y n [f;g,h] 3 L, R 4 Insets 2 =1 1 e n [ f ; 1, h; 4; λ] 4 ( f, h 1) L,R 5 Tailed insets 2 =1 2 e n [ f ; 1, h; 5; λ, µ] ( f 1, 1) (1,h 2) R 6 Banners 2 =1 3 e n [ f ; 1, h; 6; λ] (2, h 3) R 7 Nooks 2 =1 3 e n [ f ; 1, h; 7; λ] ( f 2, 2) R 8 Swivels 2 =1 =2 e n [ f ; 1, 2; 8; λ] ( f 1, 4) 5 R 2 9 Tailed swivels 3 =1 =2 e n [ f ; 1, 2; 9; λ, µ] ( f 3, 3) (2,1) R 2 10 Tagged swivels 4 =1 =2 e n [ f ; 1, 2; 10; λ] ( f 4, 4) 11 Swivel shifteds 4 =1 =2 e n [ f ; 1, 2; 11; µ] ( f 3, f + 1) 6 R 12 Pumps 3 =1 =2 e n [ f ; 1, 2; 12; λ] ( f 3, 2) 13 Tailed pumps 3 =1 =2 e n [ f ; 1, 2; 13; λ] ( f 3, 1) 14 Near bats 3 =1 =2 e n [ f ; 1, 2; 14] 15 Bat =3 =1 =2 e 7 [3; 1, 2; 15] 1 In Class 2, the shifted shape µ must contain the element depicted with the boxed square. 2 In Classes 2, 8, and 9, these elements are acyclic only when the elements depicted with the hollow triangles do not exist. 3 In Class 3, the name e n [1; g, h] is used if f = 1. 4 In Class 4, the name d n [ f ; 1, 1] is used if h = 1. 5 In Class 8, the shape λ must contain the two elements depicted with solid squares. 6 In Class 11, the shape µ must contain the four elements depicted with solid squares.
74 PROCTOR Figure 5.1. Figure 5.2. Figure 5.3.
DYNKIN DIAGRAM CLASSIFICATION 75 Figure 5.4. Figure 5.5. Figure 5.6.
76 PROCTOR Figure 5.7. Figure 5.8. Figure 5.9.
DYNKIN DIAGRAM CLASSIFICATION 77 Figure 5.10. Figure 5.11. Figure 5.12.
78 PROCTOR Figure 5.13. Figure 5.14. Figure 5.15.
DYNKIN DIAGRAM CLASSIFICATION 79 Define the top tree poset T N = T N ( f ; g, h) to be the filter of M N ( f ; g, h) consisting of all large solid dots. Also define the top filter poset U N = U N ( f ; g, h) to be the filter of M N ( f ; g, h) consisting of all large solid dots, circled dots, and boxed squares. For each N, it is obvious that T N ( f ; g, h) is the top tree of M N ( f ; g, h) and that T N ( f ; g, h) = Y ( f ; g, h). It is also obvious that U N ( f ; g, h) is the top filter of M N ( f ; g, h), i.e., that U N ( f ; g, h) = U( f ; g, h), as defined in Section 6. The elements of L N beyond U N consist of the solid dots and the solid squares. The optional poset O N = O N ( f ; g, h) is the ideal of M N consisting of all hollow dots and hollow triangles, i.e., the complement of L N in M N. The optional posets λ and µ listed in Table 1 are usually filters of O N.ForC 2,C 8, and C 11, the posets µ and λ are required to contain from one to four elements of L N as well for notational convenience. Such elements are depicted with boxed or solid squares. In 14 out of the 15 classes, the top tree Y ( f ; g, h) of the irreducible components is such that min[ f, g, h] 1. When viewed as a Dynkin diagram for a simply laced Kac-Moody algebra, most or all such trees are of type A, D, ore, depending upon how type E is defined. The rank of the algebra is n := f + g + h + 1. So we will write X n ( f ; g, h) instead of Y ( f ; g, h) when min[ f, g, h] 1, where X {A,D,E}. In particular, take X = A when f = 0 and take X = D when two of { f, g, h} are both equal to 1. Historically it has perhaps been required that { f, g, h} {1,2}and min[{ f, g, h} {1,2}] 2 in order to take X = E. But we will more generally require only 1 {f,g,h}and min[{ f, g, h} {1}] 2 in order to regard Y ( f ; g, h) as a Dynkin diagram of general type E. For the fifteenth class C 3, the only restrictions are f 1 and h g 2. There we use the letter Y rather than A, D, orewhen f 2. So each of the top trees appearing in figure 5 can be denoted X n ( f ; g, h), where X {A,D,E,Y}and n 3. Building upon this Dynkin diagram notation, we now introduce a name for each possible irreducible component P.IfPhas top tree X n [ f ; g, h], then it is assigned a name roughly of the form x n [ f ; g, h; N; λ, µ]. (The lower case x instead of the upper case X continues the convention of [8, 9] of using the lower case letter for the poset of join irreducibles P and the upper case letter for the distributive lattice J(P). This lattice J(P) is a Bruhat order, and in future papers we will denote the Bruhat order corresponding to the irreducible component x n [ f ; g, h; N; λ, µ] =: P by X n [ f ; g, h; N; λ, µ] = J(P).) If it is not determined by x, f, g, and h, then the number N of the class of which P is a member is displayed. Finally, the parameters λ and µ denote filters of O N (plus possibly a few elements of L N ) which determine the particular irreducible component P. Table 1 specifies which of the two minimal elements of T N are acyclic: Here the presence of L means that the element a g of figure 4 is acyclic, and R indicates that the element d h of that figure is acyclic. For C 2 (respectively, C 8 and C 9 ), the element a g (respectively, d h ) is not acyclic when the element of O N depicted with the hollow triangle is present. Combining the knowledge of the irreducible components and their acyclic elements with the procedure given at the end of Section 4 enables one to generate all connected d-complete posets with a given top tree. Theorem If P is an irreducible component, then it is described in exactly one of the lemmas below, for some 1 N 15. The proof of the theorem is provided by the narrative of Sections 9 13: As we proceed, we explain how the wordings of the definitions of the classes together with various local
80 PROCTOR structural conditions imply that no irreducible components are being missed. The conditions on f, g, and h which appear in Table 1 are incorporated into the definitions of the classes C N as we proceed. Lemma 7 N, 1 N 15. Let f, g, and h be fixed parameter values allowed by the Nth row of Table 1. Then the poset M N ( f ; g, h) is an irreducible component. There exist no extensions of M N ( f ; g, h) to a larger irreducible component. A poset P is in the class of irreducible components C N and has top tree Y ( f ; g, h) if and only if P contains L N ( f ; g, h) and is a filter of M N ( f ; g, h). All such filters of M N ( f ; g, h) are distinct. Specification of posets λ and/or µ from the possibilities listed in Table 1 corresponds to choosing one member of C N. The acyclic elements for each P C n are listed in Table 1. The least routine aspects of each of these 15 lemmas will be confirmed in Sections 9 13 following the definition of the corresponding class of irreducible components. Our extension arguments will imply that each M N is a slant irreducible d-complete poset. For each class we will leave several routine verifications to the reader. The non-existence of slant irreducible extensions of M N will follow from Lemma 8.A. If P is a filter of M N ( f ; g, h), then it is d-complete. If it contains L N ( f ; g, h) as well, then it can be seen that it is slant irreducible and has top tree Y ( f ; g, h). Verifying that any filter P of M N which contains L N satisfies the other particular defining conditions of C N for each N will be left to the reader. The key step is the converse: Suppose that P is in the class C N of irreducible components and has top tree Y ( f ; g, h). We will argue that P must then be a filter of M N. (It will be obvious that if P C N, then P L N.) The precise specification of members of C N with λ and/or µ will be performed as we proceed. A consequence of this specification will be the fact that the members of each class for fixed values of f, g, and h are distinct. The convention g h guarantees that members of the same class for different parameter values will never coincide. The reader may confirm the list of acyclic elements for each case using Proposition 6.B. 8. Extending d-complete posets Let P 0 be a fixed irreducible component. It will begin with a top tree Y ( f ; g, h) and a top filter U( f ; g, h) for some values of f 0, g 1, and h g. If we fix these values for f, g, and h and exhaustively list all possible irreducible components P which begin with U( f ; g, h), our fixed P 0 will eventually appear in the list of possible P s which we generate. In this section we establish the mechanics which will be used for this process in Sections 9 13. Let F be a d-complete poset. A d-complete poset P is an extension of F if F is a filter of P. An element x F isa1-extension with respect to F if F {x}is d-complete. A dangle extension x of F is a 1-extension of F such that w is covered by one element in F. A wedge extension of F is a 1-extension of F such that w is covered by two elements in F. Since elements of d-complete posets are never covered by three or more elements, every 1-extension of F is either a dangle extension or a wedge extension. The following lemma follows immediately from the definition of d-complete poset.
DYNKIN DIAGRAM CLASSIFICATION 81 Lemma A Let F be a d-complete poset with top tree T. 1. Suppose that x F is covered by only y F, and y T. Then x is a dangle extension for F if and only if it is the bottom element of some d k -interval [x, z] for some z F and some k 4 such that z covers no element in F other than one element of [x, z], the element y is not in the neck of a d k -interval, and any element in F covered by y must be covered by at least one other element besides y. 2. Suppose that w F is covered by x, y F. Then w is a wedge extension for F if and only if there exists some z F which covers only x and y, there is no element in F which is covered by both x and y, and neither x nor y is in the neck of a d k -interval for any k 3. We will start with F 0 = U( f ; g, h) and repeatedly do the following procedure: For each i 1 list all possible ways of extending each F i 1 found at the previous stage by one element meeting the requirements of the lemma. In order to focus on one set of values for f, g, and h at a time, we will not consider 1-extensions of F i 1 which extend its top tree. (These would be dangles beneath a g or d h.) So each F i produced will be an irreducible component with top filter U( f ; g, h). Let P 0 be a fixed irreducible component with top filter U( f ; g, h). Set F 0 := U( f ; g, h) and F t := P 0, where t := P 0 U(f;g,h). Let F 0 F 1 F t be any sequence of filters starting with U( f ; g, h) and increasing to P 0 one element at a time. Since all of these posets are connected, by F4 we see that each x i := F i F i 1 for 1 i t is either a dangle or a wedge extension with respect to F i 1. So repeated adjunctions of dangle and wedge extensions beginning with U( f ; g, h) are guaranteed to eventually produce P 0. Suppose that a dangle (respectively, wedge) extension is adjoined at some stage. Then in that extension and in all later extensions, that element will be covered by exactly one (respectively, two) elements. One quickly tires of considering all possible orders in which 1-extensions may be adjoined. Let x and y be two 1-extensions with respect to a d-complete poset F. Although it will be relatively rare, it is possible for y to fail to be a 1-extension with respect to F {x}. And then x will fail to be a 1-extension with respect to F {y}. It can be seen that this situation will arise only when there is some element z F which covers a 1-extension x F and z is such that [y, z] isad k -interval in F {y}for some y F and some k 3. If this cautionary note is ignored, then any violation of a d-complete requirement created by adjoining both elements will remain a violation if further elements are adjoined. Hence, any such error would eventually be detected by noting that one of our maximal irreducible components M N defined in Section 7 is not d-complete. So we can be casual concerning the order in which 1-extensions are adjoined. In addition, If x F is not a 1-extension of F, then it will never be a 1-extension with respect to any extensions of F. So once we rule out a conceivable 1-extension, we do not need to keep checking on it as a possibility. Let F be a d-complete poset. The following local situation depicted in figure 6 will often arise in Sections 9 13 when we are seeking all possible extensions of F: Suppose that for some s 1 and some t 1 we have elements a s a 1 z and b t b 1 z such that a i covers only a i+1 for 1 i < s and such that b j covers only b j+1 for 1 j < t. Also suppose that z covers only a 1 and b 1 and that a s and b t are
82 PROCTOR Figure 6. minimal in F. Finally, assume that no a i, 1 i s, and no b j, 1 j t, is a neck element in any d k -interval, for any k 3. Let F M be the poset formed by successively adjoining st wedges in a grid pattern in the region between the chains a s a 1 and b t b 1 below F. It is clear that F M will be d-complete, as will any filter P of F M which contains F. (Such posets P are obtained by successively adjoining some shape λ (s, t) of wedges below F.) What about other 1-extensions to F or a successor poset which are below z? First, some terminology. A dangle extension of F beneath an a i,ab j,orzwill be called a first generation dangle. Any wedge extension of F or of a later F which is beneath an a i and/or a b j and which has both parents below z will be called a first generation wedge. Later wedges both of whose parents are beneath z will be called higher generation wedges. Wedges which have one parent which is below z and one parent which is not below z will be called outer wedges. Any dangle extension of a later F which is beneath a first generation wedge will be called a second generation dangle. In any poset P, an element x is an only child if it is covered by some element of P which covers only x. Here is the procedure we will follow when listing all possible extensions of F below z. First note all first generation dangles. If one of these is used, then exit this procedure and consider that situation separately. Similarly, note all outer wedges and consider implementing them at another time. By D1 and D3 it can be seen that the only possible first or higher generation wedges must be the obvious grid filling wedges noted above. After choosing some first generation wedges, look for possible second generation dangles using the following result: Lemma B Let u be a first generation wedge extension. It can have one second generation dangle q beneath it if and only if the diamond top v corresponding to the diamond bottom u is an only child with respect to an element other than an a i orab j. There are no dangle extensions beneath higher generation wedges.
DYNKIN DIAGRAM CLASSIFICATION 83 Proof: Let y be such that y covers only v. Then [q, y] will be a d 4 -interval, and adjoining one dangle q beneath u creates no d-complete violations. If no such y exists, then [q,v] will be an incomplete d 4 -interval. Let u be a higher generation wedge extension and let v be the corresponding diamond top. Then v is an earlier wedge extension which we adjoined, and none of these which became diamond tops by some later obvious wedge extensions are only children. So as above, the element u can have no dangles beneath it. Outer wedge possibilities will rarely arise. By D1, the two parents must already be covered by a mutual element. This situation will arise only when the parent below z is one of the a i or b j and the grandparent is not below z. Whenever the situation of figure 6 arises in Sections 9 13, we will assume that the reader will assist us in performing the procedure described above: Note all first generation dangles and outer wedges for future consideration. Add some first generation wedges. Using Lemma B, note all second generation dangles for future consideration. Add some higher generation wedges to complete an adjoined shape λ (s, t) of wedges between the two chains. Lemma B excuses us from checking for later dangles. Altogether, this routine will be referred to as the Grid Filling Procedure, or GFP. From Sections 5 and 6, recall that any irreducible component P has a top tree Y ( f ; g, h) for some f 0 and h g 0 and contains the top filter U( f ; g, h). The classes C 1, C 2, and C 3 are defined to consist of all irreducible components having certain values for f, g, and h. To describe these classes we need to generate all extensions of U( f ; g, h). If N 4, the class C N is defined to consist of all irreducible components which contain a specified poset V as a filter, which have the same top tree as V, and which satisfy certain other conditions. Then we need to generate all extensions of V which satisfy the specified conditions. By Proposition 6.C, at the beginning of the extension process we only need to consider extensions beneath the elements b 2,...,b g, c 2,...,c h,and t 1,...,t f. 9. Classes 1 3: Shapes, shifted shapes, and birds In this section we define the three simplest classes of irreducible components and confirm the corresponding Lemma 7.N s. If the top tree of an irreducible component P is T, let f, g, and h be such that T = Y ( f ; g, h). Recall that f 0, g 1, and h g in general. To be thorough, in the definition of each class we will restate all known constraints on f, g, and h. Either f = 0or f>0. Let C 1 consist of all irreducible components for which f = 0, g 1, h g. To prove Lemma 7.1, note that the top filter U(0; g, h) for any P C 1 is as shown in figure 7. By Proposition 6.C, all extensions will be beneath e 3. Following GFP, it can be seen that the only possible extensions P of U(0; g, h) are the filters of M 1 which contain L 1 = U(0; g, h). Each such filter is determined by its intersection with O 1 ; such intersections are elements λ of (g 1, h 1). (Note that each poset a n [0; g, h; λ] C 1 is a shape whose first two columns are of length g + 1 and whose first two rows are of length h + 1.)